According to one theory, ice cream headaches are caused by an increase in blood volume of the anterior cerebral artery:

Another theory into the cause of ice-cream headaches is explained by increased blood flow to the brain through the anterior cerebral artery, which supplies oxygenated blood to most medial portions of the frontal lobes and superior medial parietal lobes. This increase in blood volume and resulting increase in size in this artery is thought to bring on the pain associated with an ice-cream headache.

Does repeatedly experiencing this phenomenon put someone at higher risk for some sort of cerebral vascular accident, or is it not any more harmful than a small increase in blood pressure that one might experience under exertion, despite the painful nature of the headache?

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I do not have a source for the following as it was taught to me in neuroanatomy classes, the explanation given for brain freeze is that it is referred pain through the trigeminal nerve. The meninges (the tissue lining the central nervous system) is innervated by the trigeminal nerve in the head. Therefore the pain feels as though it is like a headache from meningeal irritation but is in fact benign.

Given that the mechanism is through referred pain rather than any vascular event it should not put you at any increased risk.

This being said if you experience repeated headaches and are unsure of the cause it would be wise to consult your doctor.